Let $k=\Bbb F_q$ be a finite field, $\mathcal{G}$ be a reductive group over $k$, denote $m(G)$ by the minimal dimension of faithful representation of $G=\mathcal{G}(\Bbb F_q)$. Do we know the value $m(G(F_q))$ explicitly at least in the $GL_n$ case?

For example, by classification of irreducible complex representations of $GL_2(\Bbb F_q)$ (assume $q$ is not power of $2$), we know $m(GL_2(\Bbb F_q))=q-1$ (a suitable cuspidal representation of dimension $q-1$ is faithful ).